Project 4

You may work in a group of 4 people maximum, or work alone.

Problem 1
p. 73 Exercise 2.1.13 and some additional directions:
Part A:
By a patch, the book means local coordinates. In order to examine
the role of u and v, hold one constant and think about what kind of curve
the other gives. This is an extrinsic definition. While it is not listed,
R is the distance away from the origin.

Part B: In addition, compute
Xu x Xv and show that it is never the 0 vector. What does this tell you
about regularity and the existence of that tangent plane?

Part C: Compute F=Xu · Xv. Interpret your result,
ie what does this tell you about the relationship between Xu and Xv.

For parts B and C, you may wish to use the following commands and
procedures in Maple:
Xu := [diff(X[1],u),diff(X[2],u),diff(X[3],u)];
Xv := [diff(X[1],v),diff(X[2],v),diff(X[3],v)];

Part D:
Open up the Maple demo that is accessible from the main web page,
and input the torus instead of the sphere.
Choose specific values for r and R.
Explore and find two different curves on the your specific (r,R) torus -
one that is a geodesic and one that is not.
For each curve:
Part D(1):
Provide your new values for each of the following (these are the commands
I used for the sphere):
g := (x,y) -> [cos(x)*sin(y), sin(x)*sin(y), cos(y)]:
a1:=0: a2:=Pi: b1:=0: b2:=Pi:
c1 := 1: c2 := 3:
Point := 2:
f1:= (t) -> t:
f2:= (t) -> 1:
Part D(2): Sketch by-hand or print out a
picture of the curvatures and the torus.
Part D(3): Discuss your curves
from an intrinsic point of view - ie why is the
geodesic a geodesic, and why is the other curve not "straight." You
should refer to an intrinsic argument - ie symmetries, the ribbon test,
and/or a covering space argument.

Part E: You have already calculated F. Calculate E and G.
Set up a double integral that uses the metric coefficients
that could be used to find the surface area of the torus.
Explain your limits of integration.

Part B: Next examine
the metric form (ds/dt)2 and use this to discuss whether
the Pythagorean theorem holds on this surface.

Part C: The flat torus can be obtained by taking a
square and identifying the edges straight across (top to bottom and
separately left to right). So a covering space would be infinitely squares
next to each other which are exact copies of each other:

How many geodesics join two points? Explain and draw pictures
in the covering space.

Part D: Can a geodesic on the flat torus ever intersect
itself? Explain and draw pictures in the covering space.